Two-stage stochastic optimization problems take the form (REF ) with \(f_0(x)\)  being the first-stage cost of decision \(x\)  and \(f_i(x)\)  specifying the second-stage cost in scenario \(i\) , which then occurs with probability \(p_i\) . The second-stage cost could equal infinity in the absence of complete recourse, which leaves open the possibility that \(f_i(x) = \infty \) ; see for example [1]} and [2]}. The latter produces an induced constraint in (REF ) when \(p_i >0\) . However, if \(p_i = 0\)  because the underlying data point is an outlier or represents an irrelevant situation, then \(p_if_i(x) = 0\)  and \(x\)  might even be a minimizer in (REF ). An alternative probability \(p_i^\nu > 0\) , regardless of how close it is to the actual probability \(p_i = 0\) , would still lead to the conclusion that \(x\)  is infeasible because \(p_i^\nu f_i(x) = \infty \) .
